ABSTRACT

This chapter aims to construct quasi-interpolants in terms of refinement sequences directly. It shows that the results are consistent with the theory of cardinal B-splines. For this purpose, it is necessary to give a precise definition of sum rules, by specifying their orders. The chapter employs the notion of “commutators” and develops an extension of “Marsden’s identity” to formulate and compute moment sequences that arise from sum rules of the refinement sequences of higher orders. These moment sequences are instrumental to the formulation of our quasi-interpolation operators. It characterizes the sum-rule order m in terms of the polynomial factor of the Laurent polynomial symbol of the refinement sequence.